Annual profit p is the product of total annual sales s and

Question

Question: Annual profit (P) is the product of total annual sales (S) and profit per unit sold (X); that is, P = X · S. It is desired to know the probability distribution of the random variable P when X and S have the following assumed probability mass functions (X and S are independent):

X (Unit Profit)                                                 S (Annual Sales)

Value                     Probability                  Value                        Probability

0                                0.05                       5                                 0.4

1                                0.45                       6                                 0.2

2                                0.30                       8                                  0.4

3                                0.20

What are the mean, variance, and standard deviation of the probability distribution for annual profit, P?